Projective equivalence of affine connections (${\bar{\Gamma }}^a{}_{bc}-{\Gamma }^a{}_{bc}={\delta }_b^a{V}_c+{\delta }_c^a{V}_b$)

Two torsion-free connections are projectively equivalent if they have the same unparametrized geodesics. Their difference of affine connections is a tensor $S$. A change of parameter adds only tangential acceleration, so equality of curves requires $S(v,v)$ to be parallel to $v$ for every tangent vector. The symmetric bilinear diagonal-parallel lemma gives exactly the displayed condition. Conversely the geodesic reparametrization under projective equivalence absorbs this tangential term. Agreement with the same affine parameters instead forces $S=0$.

Let $\tau$ be an affine parameter for a geodesic of $\nabla$ and suppose ${\bar{\nabla }}_{X}Y-{\nabla }_{X}Y=V(X)Y+V(Y)X$. Then ${\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }=2V(\dot{\gamma })\dot{\gamma }$. For the new velocity $d\gamma /dt=\dot{\gamma }/t^{\prime }$, the acceleration is $(t^{\prime }{)}^{-2}[{\bar{\nabla }}_{\dot{\gamma }}\dot{\gamma }-(t^{\prime \prime }/t^{\prime })\dot{\gamma }]$, which vanishes under the displayed differential equation. Its nonzero solution $t^{\prime }=C\exp (2\int V(\dot{\gamma })d\tau )$ changes the parameter while preserving the curve.

For two projectively equivalent connections that are Levi-Civita connections, contraction of their difference gives $(n+1)V_c={\bar{\Gamma }}^a{}_{ac}-{\Gamma }^a{}_{ac}$. The Jacobi formula gives ${\Gamma }^a{}_{ac}={\partial }_c\log \sqrt{|\det g|}$. Subtraction proves the displayed expression. The two determinant densities transform by the same coordinate Jacobian factor, so their ratio is a scalar and its derivative a covector. The formula is a necessary consequence of projective equivalence, not a test asserting that arbitrary metric tensors have identical geodesics.